In selected null simulations, Ville’s bound overstated the rate at which a martingale crossed a threshold by as much as 2.4 times. The paper’s overshoot-based identity stayed within the observed binomial interval at 41 of 44 levels. Its median relative gap was 0.009 in the WDBC stream and 0.028 in the Pima stream, compared with 0.402 and 0.243 for Ville’s bound.
The tests used a sequential likelihood ratio for a held-out classifier against a label-marginal null, with 60,000 independently redrawn-label null replicates for each stream. The WDBC and Pima horizons were 285 and 384 time steps.
A framework for the missing pieces
The arXiv preprint asks how information discarded when martingale concentration inequalities drop nonnegative terms can be represented and measured on trajectory and random-time spaces. Its main method places a mixed-coincidence variational identity on filtered path spaces that track information step by step, then uses the relative-entropy chain rule to resolve the path-level residual into conditional terms at each step.
The framework describes the discarded slack in three geometries: crossing for Ville and pooled tests, Gibbs tilts for Azuma–Hoeffding and PAC-Bayes, and dominating certificates for the Lp maximal bound.
When the clock is random
One part examines what happens when an e-process is evaluated at a random time. The path-time calculus factors that time into a hazard clock and an anticipation martingale, and treats the extra factor as a peeking penalty. The penalty is nonpositive at stopping times, zero at pseudo-stopping times for uniformly integrable martingales, and potentially unbounded for look-ahead times.
Discrete paths expose corrections
A separate check used a symmetric ±1 walk rather than a dataset. For the quoted curved boundary, the nominal crossing level was 0.3935; the reported quadrature residual was 0.1343 and the overshoot was 0.08571. The resulting crossing probability was 0.4421, or 12.35% above nominal. Closed-form and simulated event identities agreed on every reported simulated path across 1.25 million paths and five horizons.
In another check, the first-passage ratio at level 5 barely changed when the horizon was expanded by a factor of 40. It remained 0.8416, 0.7504 and 0.6326 at tilts 0.3, 0.5 and 0.8 across 300,000 nested-prefix paths. At the same level, the reported Gaussian deficit was 0.16–0.37 at a unit step and 0.0055–0.0148 at dt = 1/1024.
The examples turn to model choices
One machine-learning example showed how the reported certificate depended on the reference choice. In a frozen CIFAR ResNet-20 evaluation, the origin-anchored certificate was 41.7 against a held-out risk of 0.078. With an anchor at the trained weights, the certificate was 0.153 against the same risk. The reported tightening was 273-fold across five seeds.
For pooled tests, the paper gives a closed-form crossing probability for a geometric mixture of multiple test martingales. It writes the deficit as three parts: disagreement shed before crossing, mass that never reaches the threshold, and crossing overshoot.
The paired pooling check used two disjoint pools of 12 checkpoints, ensemble sizes of 1, 2, 4 and 8, a 5,000-image ImageNet-100 validation stream, and 20 stream orderings per cell across 240 cells. Mixture mass fell to 0.739, 0.590 and 0.514 at ensemble sizes 2, 4 and 8. At matched accuracy, the reported excess differences were +0.086 at size 4, with a 95% confidence interval of 0.047 to 0.124, and +0.123 at size 8, with an interval of 0.095 to 0.151.
Where the evidence stops
These are mathematical identities under stated filtration, integrability, martingale, boundary and stopping assumptions, not universal correction factors. The random-time result allows an unbounded penalty for look-ahead times, while the pooled identity is stated for geometric mixtures of nonnegative test martingales. The numerical checks remain tied to the selected processes, datasets and model configurations described in the paper.
The deepest crossing levels in the WDBC and Pima checks were sparse and carried wider sampling uncertainty. In the curved-boundary experiment, the asymptotic level was not reached at finite simulated horizons.
The deep-net certificate was evaluated at its maximum over the run, a look-ahead time, using a frozen CIFAR ResNet-20 with 92.2% held-out accuracy and a 650-weight last-linear head.
The work is an arXiv preprint dated 20 Aug 2026. The supplied text reports no funding statement; its acknowledgment discloses large-language-model use and assigns sole responsibility to the author.
Paper data and sources
Original title: Information on trajectories: martingales and random times
Authors: Akshay Balsubramani
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text